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user64494
user64494
4:37 AM
No, this is not any trolling. This is a serious matter which requiers a serious answer. Unfortunately, there is no satisfactory answer from Henrik Schumacher and others.
 
 
7 hours later…
Henrik Schumacher
Henrik Schumacher
11:14 AM
@user64494 Before, I tried to convince you that `DiracDelta[x] DiracDelta[y]` is meaningful by referring to Fubini's theorem (without mentioning Fubini's name).
Maybe you find the following construction more convincing:
The tensor product $\mathcal{D}(\mathbb{R}) \otimes \mathcal{D}(\mathbb{R})$ is dense in $\mathcal{D}(\mathbb{R}^2)$, hence we may write $\mathcal{D}(\mathbb{R}^2) = \mathcal{D}(\mathbb{R}) \overline{\otimes} \mathcal{D}(\mathbb{R})$.
We can define $\delta \otimes \delta \in (\mathcal{D}(\mathbb{R}) \otimes \mathcal{D}(\mathbb{R}))'$ on simple vectors $\varphi \otimes \psi$ by $\langle \delta \otimes \delta, \varphi \otimes \psi\rangle := \langle \delta ,\varphi \rangle \cdot \langle \delta , \psi \rangle$. Extending it continuously leads to a distribution $\delta \otimes \delta \in \mathcal{D}'(\mathbb{R})$.
When inserting arguments, we obtain $(\varphi \otimes \psi)(x,y) = \varphi(x) \, \psi(y)$. In order to write that into an integral over $\mathbb{R}^2$, one has to indicate which $\delta$ is in which variable.
So the notation
$$
\iint_{\mathbb{R}^2} \delta(x) \, \delta(y) \, \varphi(x) \, \psi(y) \, \mathrm{d} x\, \mathrm{d} y = \langle \delta \otimes \delta, \varphi \otimes \psi\rangle
$$
or more general
$$
\iint_{\mathbb{R}^2} \delta(x) \, \delta(y) \, f(x,y)\, \mathrm{d} x\, \mathrm{d} y = \langle \delta \otimes \delta, f\rangle,
$$
while not meeting your high standards of aesthetics, can be still interpreted in a meaningfull way. Since Mathematica has currently no other way of formalizing the pairing $\rangle \cdot, \cdot \langle$ between distributions and test functions, it simply has to use
@user64494 Finally a remark towards your attitude: A single thing not working exactly as you desire in a CAS does not imply that the whole system were useless or meaningless. And by no means, it implies that their users have no dignity. So, please treat them like they deserve it as human beings.
 
 
4 hours later…
AccidentalFourierTransform
AccidentalFourierTransform
2:59 PM
@HenrikSchumacher You are wasting your time: user64494 has a history of commenting on every post that mentions distributions (check their comment history). They always claim that the OP misunderstands distributions which, unfortunately, is never the case: it is user64494 who does not understand distributions.
2
this behaviour is not only very annoying and condescending, but also somewhat dangerous: people unfamiliar with user64494's behaviour might think that he/she is right, and OP is wrong.
(and there is always a downvote from user64494 on correct/good posts, which is also problematic to some extent)
and although user64494 has been corrected repeatedly by many different users, he/she does not seem to reevaluate his/her understanding of distributions, or to change his/her behaviour
I believe something should be done about this disruptive behaviour, but for now, the best thing we can do is to ignore him/her
 
Henrik Schumacher
Henrik Schumacher
@AccidentalFourierTransform Thanks for this info. I've come to the same conclusion...
 

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